Algebra Examples

$\frac{3}{2} u^{2} v^{4} - 2 \frac{1}{2} u^{3} v^{5} + \frac{1}{2} u^{4} v^{2}$

Step 1

Convert $2 \frac{1}{2}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

$\frac{3}{2} \cdot (u^{2} v^{4}) - (2 + \frac{1}{2}) u^{3} v^{5} + \frac{1}{2} \cdot (u^{4} v^{2})$

Step 1.2

Add $2$ and $\frac{1}{2}$ .

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Step 1.2.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{2} \cdot (u^{2} v^{4}) - (2 \cdot \frac{2}{2} + \frac{1}{2}) u^{3} v^{5} + \frac{1}{2} \cdot (u^{4} v^{2})$

Step 1.2.2

Combine $2$ and $\frac{2}{2}$ .

$\frac{3}{2} \cdot (u^{2} v^{4}) - (\frac{2 \cdot 2}{2} + \frac{1}{2}) u^{3} v^{5} + \frac{1}{2} \cdot (u^{4} v^{2})$

Step 1.2.3

Combine the numerators over the common denominator.

$\frac{3}{2} \cdot (u^{2} v^{4}) - \frac{2 \cdot 2 + 1}{2} u^{3} v^{5} + \frac{1}{2} \cdot (u^{4} v^{2})$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $2$ by $2$ .

$\frac{3}{2} \cdot (u^{2} v^{4}) - \frac{4 + 1}{2} u^{3} v^{5} + \frac{1}{2} \cdot (u^{4} v^{2})$

Step 1.2.4.2

Add $4$ and $1$ .

$\frac{3}{2} \cdot (u^{2} v^{4}) - \frac{5}{2} u^{3} v^{5} + \frac{1}{2} \cdot (u^{4} v^{2})$

Step 2

Simplify each term.

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Step 2.1

Multiply $\frac{3}{2} (u^{2} v^{4})$ .

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Step 2.1.1

Combine $u^{2}$ and $\frac{3}{2}$ .

$\frac{u^{2} \cdot 3}{2} v^{4} - \frac{5}{2} u^{3} v^{5} + \frac{1}{2} \cdot (u^{4} v^{2})$

Step 2.1.2

Combine $\frac{u^{2} \cdot 3}{2}$ and $v^{4}$ .

$\frac{u^{2} \cdot 3 v^{4}}{2} - \frac{5}{2} u^{3} v^{5} + \frac{1}{2} \cdot (u^{4} v^{2})$

Step 2.2

Move $3$ to the left of $u^{2}$ .

$\frac{3 \cdot u^{2} v^{4}}{2} - \frac{5}{2} u^{3} v^{5} + \frac{1}{2} \cdot (u^{4} v^{2})$

Step 2.3

Combine $u^{3}$ and $\frac{5}{2}$ .

$\frac{3 u^{2} v^{4}}{2} - \frac{u^{3} \cdot 5}{2} v^{5} + \frac{1}{2} \cdot (u^{4} v^{2})$

Step 2.4

Move $5$ to the left of $u^{3}$ .

$\frac{3 u^{2} v^{4}}{2} - \frac{5 \cdot u^{3}}{2} v^{5} + \frac{1}{2} \cdot (u^{4} v^{2})$

Step 2.5

Combine $v^{5}$ and $\frac{5 u^{3}}{2}$ .

$\frac{3 u^{2} v^{4}}{2} - \frac{v^{5} (5 u^{3})}{2} + \frac{1}{2} \cdot (u^{4} v^{2})$

Step 2.6

Move $5$ to the left of $v^{5}$ .

$\frac{3 u^{2} v^{4}}{2} - \frac{5 \cdot v^{5} u^{3}}{2} + \frac{1}{2} \cdot (u^{4} v^{2})$

Step 2.7

Multiply $\frac{1}{2} (u^{4} v^{2})$ .

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Step 2.7.1

Combine $u^{4}$ and $\frac{1}{2}$ .

$\frac{3 u^{2} v^{4}}{2} - \frac{5 v^{5} u^{3}}{2} + \frac{u^{4}}{2} v^{2}$

Step 2.7.2

Combine $\frac{u^{4}}{2}$ and $v^{2}$ .

$\frac{3 u^{2} v^{4}}{2} - \frac{5 v^{5} u^{3}}{2} + \frac{u^{4} v^{2}}{2}$

Step 3

Simplify terms.

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Step 3.1

Combine the numerators over the common denominator.

$\frac{3 u^{2} v^{4} - 5 v^{5} u^{3}}{2} + \frac{u^{4} v^{2}}{2}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{3 u^{2} v^{4} - 5 v^{5} u^{3} + u^{4} v^{2}}{2}$

Step 3.3

Factor $u^{2} v^{2}$ out of $3 u^{2} v^{4} - 5 v^{5} u^{3} + u^{4} v^{2}$ .

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Step 3.3.1

Factor $u^{2} v^{2}$ out of $3 u^{2} v^{4}$ .

$\frac{u^{2} v^{2} (3 v^{2}) - 5 v^{5} u^{3} + u^{4} v^{2}}{2}$

Step 3.3.2

Factor $u^{2} v^{2}$ out of $- 5 v^{5} u^{3}$ .

$\frac{u^{2} v^{2} (3 v^{2}) + u^{2} v^{2} (- 5 v^{3} u) + u^{4} v^{2}}{2}$

Step 3.3.3

Factor $u^{2} v^{2}$ out of $u^{4} v^{2}$ .

$\frac{u^{2} v^{2} (3 v^{2}) + u^{2} v^{2} (- 5 v^{3} u) + u^{2} v^{2} (u^{2})}{2}$

Step 3.3.4

Factor $u^{2} v^{2}$ out of $u^{2} v^{2} (3 v^{2}) + u^{2} v^{2} (- 5 v^{3} u)$ .

$\frac{u^{2} v^{2} (3 v^{2} - 5 v^{3} u) + u^{2} v^{2} (u^{2})}{2}$

Step 3.3.5

Factor $u^{2} v^{2}$ out of $u^{2} v^{2} (3 v^{2} - 5 v^{3} u) + u^{2} v^{2} (u^{2})$ .

$\frac{u^{2} v^{2} (3 v^{2} - 5 v^{3} u + u^{2})}{2}$